There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group , including the following statements: the ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weights four and six, denoted by and ; the smallest-weight cusp form has weight twelve and can be written as a polynomial in and ; and the Hauptmodul can be written as a multiple of divided by . The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups with square-free level , which are related to ‘Monstrous moonshine conjectures’. Certain aspects of our results are generated from extensive computer analysis; as a result, many of the space-consuming results are made available on a publicly accessible web site. However, we do present in this article specific results for certain low-level groups.
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